Multi-Story Building Model for Efficient IoT Network Design

Abstract

This article presents a new network model for IoT that is based on a multi-story building structure. The model locates network nodes in a regular, cubic lattice-like structure, resulting in an equation for the signal-to-noise ratio (SNR). The study also determines the relationship between traffic density, network density, and SNR. In addition, the article explores the potential of percolation theory in characterizing network functionality. The findings offer a new approach to network design and planning, allowing for selecting a network topology that meets criteria and requirements while ensuring connectivity and improving efficiency. The developed analytical apparatus provides valuable insights into the properties of the network and its applicability to specific conditions.

1. Introduction

The development of the Internet of Things (IoT) has led to the formation of high-density networks [1]. In several works, the features of those networks are shown, mainly because the network nodes have to work in conditions of relatively low signal-to-noise ratios (SNR) [2,3]. This is due to co-channel interference created by neighboring nodes that are expected to be of a large number in dense networks [4]. Low SNR increases the number of transits in the path between the source and destination nodes. This increases the e average path length and, thus, increases the communication latency.

Many existing proposals consider networks with nodes located in the same plane, i.e., networks that can be described by two-dimensional models [5,6,7]. Such networks can describe a wide range of applications [8]. These networks are located in open spaces, e.g., outdoors, or networks limited to one level, and the premises are built on one floor. However, in modern cities structure, most IoT nodes are concentrated within residential and office buildings, including smart homes, access control systems, security systems, housing, and communal systems [9,10]. The nodes of those networks can interact within the same plane, i.e., floor, and between the nodes of different planes. These networks are distributed within multi-story buildings and are generally of three-dimensional structures [11].

The structures of that networks are largely determined by the parameters of the buildings in which they are integrated. The distribution of network nodes within one plane can be described using the duty process, as indicated in [12]. With such distribution, nodes within one plane can have self-similarity properties due to properties in the structure describing rooms located on the corresponding floor of a building [12,13].

The propagation of radio signals inside a building depends significantly on its architecture and the materials used to construct walls and floors [14]. In this regard, the network model differs significantly from one structure to another due to obstacles in the form of walls and floors. Moreover, the presence of the third dimension will affect the models [12]. Thus, the network model should consider the environmental structure and properties, including obstacles [15].

To this end, this work aims to build a novel network model for IoT networks with dense deployment scenarios based on a multi-story building. The proposed model takes into account the peculiarities of its placement in three-dimensional space and the network structure. The developed model provides an expression for the SNR when arranging network nodes in a regular structure, forming a simple cubic lattice. Moreover, the SNR’s dependences on traffic density and total network nodes are determined. The theory of percolation is applied to describe the functionality of the network. The main contributions of the work are summarized as follows.

• Developing a novel network model for dense IoT networks based on a multi-story building.
• Analyzing the developed 3D-Latice structure of IoT networks.
• Analyzing the signal-to-interference noise ratio (SINR) of the developed network model.
• Analyzing the effect of traffic intensity on the SINR in the proposed model.
• Numerical evaluation of the developed network model.
• Investigating the feasibility of using percolation theory to describe the functionality of the proposed network.

The rest of the article is organized as follows. Section 2 presents the related work and the novelty of the proposed model. Section 3 presents the multi-story-based network model. Section 4 provides the numerical evaluations of the developed network model. Section 5 investigates the possibility of using percolation theory to describe the functionality of the proposed network model. Section 6 concludes the work and provides future directions.

2. Related Work

The modeling of IoT networks has attracted many researchers in recent years, especially with the new requirements of next-generation networks [16,17]. However, many developed studies considered certain IoT applications, such as healthcare and smart metering, or certain kind of IoT networks, such as LoRa [18].

In [19], the authors developed a LoRa indoor network architecture. The model investigates the LoRa Performance at 433 MHz and 868 MHz bands inside a multi-story building. The work mainly focuses on measuring signal strength indicators (RSSI), packet delivery ratio (PDR), and SNR Parameters at different floors. The study provided a comparative table of packet delivery ratios at different spreading factors. The experimental evaluation proved that the 433 MHz LoRa module achieved the strongest signal at different spreading factors, and the 868 MHz LoRa module achieved the highest percentage of received packets. The study proved that the 868 MHz LoRa network at the 10th spreading factor achieves the highest performance for a nine-story building with concrete floors. The work considered a certain type of long-range IoT network, i.e., LoRa, while the proposed work considers any IoT network. Moreover, the proposed work considers the dense deployment scenarios of IoT networks, which are not investigated in this study.

Due to the difficulty of gaining sufficient accuracy in an interior environment with low-complexity tracking technology, indoor placement remains a significant concern. In [20], the authors proposed a robust 3D indoor positioning system appropriate for indoor IoT applications. This system is built on a Bayesian network that is dependent on the strength of WiFi signals. It was evaluated experimentally in a building with pre-installed access points (APs).

In [21], the authors developed a StoryTeller, which is a technique for floor prediction in multi-story buildings based on deep learning. The proposed model used WiFi signals to generate images that are fed into a convolutional neural network (CNN). The considered CNN was trained to predict floors based on discovered patterns in visible WiFi scans. Input photos are generated so as to record the current WiFi scan independent of the AP. Also, a novel idea of virtual buildings was employed to standardize the data in order to make it building-independent. This allows StoryTeller to reuse a trained network for a brand-new building, significantly decreasing deployment costs. The results indicate that StoryTeller can estimate the user’s floor at least 98.3% of the time.

The novelty of the proposed work comes from considering all kinds of IoT networks. Also, the proposed model considers the dense deployment scenarios of IoT networks, which represents a main feature of next-generation networks.

3. Proposed Network Model

The structure of IoT networks in multi-story buildings and structures can differ significantly from flat ones, i.e., two-dimensional (2D). This is due to the presence of the third dimension, which results in inhomogeneities that occur on the path of radio signal propagation in multi-story buildings [15,22]. Such inhomogeneities are walls, inter-floor ceilings, and other building structure elements. Thus, multi-story buildings are regular structures consisting of typical elements, i.e., typical premises, and are generally self-similar fractal structures.

Within the boundaries of one room, network nodes are in the line-of-sight path at relatively short distances. The SNR can be quite high when modeled considering a network in one separate room [12]. Concrete walls and floor slabs of modern buildings introduce a significant signal attenuation, from 9 to 28 dB, according to [14], depending on the signal frequency and the type of building.

In order to analyze the functioning of the network within the entire building, we should ensure the possibility of interaction between nodes located in different rooms. However, all nodes in the same room can be considered as a single node due to the small, closed line-of-site distance between them. Let’s assume that the structure of the building consists of identical rectangular rooms, which in most cases, fit the real situations. A rectangular lattice can describe the structure of the network within the boundaries of the building, as presented in Figure 1. The nodes of the lattice are the nodes of the network and the connection edges between them [23]. Figure 2 presents a simple layout of network nodes in a multi-story building according to the proposed model.

The functioning of such a network is described by the delivery of messages from the source node, n_s, to the destination node, n_t. Such delivery is possible when there is at least one route between the source and destination nodes. In the lattice structure shown in Figure 1, there are connections, i.e., red lines, only with neighboring sites, i.e., nodes located in adjacent rooms. This assumption is valid since communications are possible over long distances. Furthermore, this assumption is justified when estimating the lower bounds of the network functionality is required.

To describe the model of the interaction of nodes in the network, we assume that the noise power level is significantly higher than the level of natural thermal noise. Thus, thermal noise is neglected in the proposed model.

The attenuation inside the premises of the considered structure can be calculated using the empirical attenuation model developed in [14], as follows.

$$L\left(d\right)=20\log f+N\log d+L_f\left(g\right)-28,$$

(1)

where d—distance (m), f—frequency (MHz), N—power loss coefficient, L_f—Loss coefficient due to signal passing through an obstacle (dB), and g—number of obstacles.

Using our previously developed model introduced in [12], we describe the analytical model of forming the interference signal for three dimensions.

$$p_{OI}=d_0{{\displaystyle \iiint }}_{V}a\left(x,y,z\right)dxdydz,$$

(2)

where d₀—interference power produced per unit volume W/m³, a—coefficient that determines the dependence of the interference power at point o on the coordinates (x,y,z) from the interference source, and v—scope of consideration.

For the proposed network model introduced in Figure 1, the dependence can be expressed in terms of the distance to the interference source, d, as follows.

$$a\left(d\right)=\frac{q}{f^2{d}^{\alpha }\mathsf{\Lambda }\left(g\right)},$$

(3)

where d—distance from the observation point to the source of interference, α—coefficient depending on the signal propagation conditions (α ≥ 2), f—signal frequency (MHz), q—constant, which has the value of 102.8 according to our previously developed model introduced in [12], Λ(g)—signal attenuation by obstacles, and g—number of obstacles.

For the proposed model, with the layout introduced in Figure 1, a discrete set of nodes is considered. Considering the discrete structure, the network can be modeled as the sum over three dimensions. The model is formulated as follows.

$$p_{OI}\left(i,j,k\right)=d_0{{\displaystyle \sum }}_{i,j,k=-\infty }^{\infty }a\left(i,j,k\right),$$

(4)

where p₀—node transmitter’s power (W). The proposed model’s lattice is not limited in each dimension, and the observation point is located at node n_0,0,0. The values of the indices allow us to indicate any lattice node and take integer values i,j,k = −∞,…,∞.

If the dimensions of the unit cell, i.e., length, width, and height, in the proposed model, introduced in Figure 2, are equal l,w,h accordingly, the distance d between the observation point and an arbitrary lattice site can be defined as follows.

$$d\left(i,j,k\right)=\sqrt{{\left(i-l\right)}^2+{\left(j-w\right)}^2+{\left(k-h\right)}^2},$$

(5)

The signal attenuation due to obstacles is calculated according to the proposed model (1), as follows.

$$\mathsf{\Lambda }\left(g\right)=Ag$$

(6)

where A—obstacle attenuation coefficient. The value of A for a concrete wall or floor slab ranges from 7.9 (9 dB) to 631.0 (28 dB). The A value recommended for residential buildings for the 2.4 GHz frequency is 10 (10 dB).

In the proposed model, the signal propagates strictly along the straight line connecting the observation point with the node n_i,j,k. Then the number of obstacles in this direction is equal to the number of intersected vertical and horizontal planes drawn through the centers of the ribs connecting the lattice nodes, i.e., the sum of the absolute values of the indices i, j, and k. Thus, Equation (6) can be modified as follows.

$$\mathsf{\Lambda }\left(i,j,k\right)=A\left(\left|i\right|+\left|j\right|+\left|k\right|\right)$$

(7)

The dependence expression introduced in Equation (3) can be rewritten using the power reduction factor. Equation (8) presents the modified dependence.

$$a\left(i,j,k\right)=\frac{q}{f^2{\left(\sqrt{{\left(il\right)}^2+{\left(jw\right)}^2+{\left(kh\right)}^2}\right)}^{\alpha }A\left(\left|i\right|+\left|j\right|+\left|k\right|\right)}$$

(8)

Considering Equations (5) and (6), Equation (2) can be written in terms of the indices of the lattice nodes. The lattice is regular and symmetric with respect to the observation center. Therefore, only positive values of the indices can be considered, i.e., 1/8 of the lattice, by introducing the corresponding factor into the expression without loss of generality. The threshold level of the power, p_m, is maintained to the receiving point, above which the signal is not perceived as interference. Then, the expression for the interference power at the observation point can be written as follows.

$$p_{OI}={\rho }_8{\displaystyle \sum }_{i,j,k=1}^{m}I\left(p_{i,j.k}\right)p_{i,j.k}$$

(9)

$$p_{i,j,k}=p_0\frac{q}{f^2{\left(\sqrt{{\left(il\right)}^2+{\left(jw\right)}^2+{\left(kh\right)}^2}\right)}^{\alpha }A\left(i+j+k\right)}w$$

(10)

$$I\left(p_{i,j,k}\right)=\left\{\begin{array}{l}{p}_{i,j,k}\hspace{1em}{p}_{i,j,k}<p_m\\ 0\hspace{1em}\hspace{1em}{p}_{i,j,k}\ge p_m\end{array}\right.\hspace{1em}w$$

(11)

where p_i,j,k—interference power generated by node i,j,k at the observation point, ρ—load intensity, m—the number of lattice nodes, and I(p_i,j,k)—indicator function that only sums the signal strengths perceived as interference.

4. Numerical Evaluation

The proposed model is evaluated using Mathcad 15 environment. This section presents the simulation setup and the numerical results.

Mathcad is a computer algebra system from the class of computer-aided design systems. It focuses on the reparation of interactive documents with calculations and visual support. Moreover, Mathcad is distinguished by ease of use and application for teamwork. Mathcad has an intuitive and easy-to-use user interface. You can use the keyboard and special toolbars to enter formulas and data.

The above analytical expressions are used to obtain point estimates and graphical dependencies. The modeling process involves implementing the proposed model using the previously described analytical terms and variables. All dependencies are presented in graphs and diagrams, simplifying the task of modeling and analyzing the results.

Figure 3 shows the dependence of the interference power level at each node of grid nodes, and the traffic intensity is calculated as follows.

$${\tilde{p}}_{OI}=10lg\left(p_{OI}/0.001\right)\mathrm{dBm}$$

(12)

As shown in Figure 3, the noise power level at the observation point increases with the number of nodes which is also evident from Equation (8). Still, its growth slows down rapidly due to a rapid increase in the attenuation of interference signals. These interferences are due to the increased communication distance and the number of obstacles in the signal propagation path.

Thus, the interference increases with the traffic density increase and the number of deployed nodes in the network. The interference significantly increases with the number of nodes until a certain level, and then with the increase of nodes, the interference is nearly constant. The traffic density is the only factor affecting the interference level at this level.

An increase in traffic intensity is directly proportional to the interference power level. The signal-to-interference noise ratio (SINR) at the observation point can be obtained as follows.

$$SINR={\tilde{p}}_{OS}-{\tilde{p}}_{OI}$$

(13)

where ${\tilde{p}}_{OS}$—signal power level at the observation point (dBm), and ${\tilde{p}}_{OI}$—interference power level at the observation point (dBm) determined according to Equation (12). The signal power level at the observation point is calculated as follows.

$${\tilde{p}}_{OS}=10\mathit{lg}\left(a\left(i,j,k\right)p_0/0.001\right)$$

(14)

The dependence of the SINR on the number of nodes and the traffic intensity is estimated in Figure 4.

Numerical results of Figure 4 indicate that, at close to real values of the number of nodes and traffic intensity, the SINR for neighboring nodes of the lattice takes a value of at least 20 dB, which in most cases allows ensuring a sufficiently high quality of the radio channel. This value of SINR is achieved for near-real situations in the case of the number of deployed nodes and the traffic intensity. For example, it is possible to operate at the maximum speed when using the IEEE 802.11ac standard with this SINR value [24,25].

Thus, the resulting model makes it possible to estimate the quality of the communication channel by estimating the SINR for network nodes. The use of the model allows us to assert that a wireless communication network can be built in a modern multi-story building with nodes in each of the rooms. The radio channel quality allows for using the maximum data transfer rates attainable for modern standards.

5. Percolation Theory and Network Performance

The percolation theory originates from the formulation of the problems of describing the physical processes of penetration (flow) of a liquid or gas through porous media, as well as the problems of describing phase transitions in various media under the influence of a factor caused by a change in the molecular structure [26,27].

In the percolation theory, various structures are considered, particularly various types of lattices used as models of the medium, e.g., crystal lattices.

The main task of the percolation theory is to describe the conditions for the “percolation”, for example, a liquid through a medium, which are sometimes called the conditions for the appearance of a percolation cluster, i.e., a set of adjacent lattice elementary cells that are a conductor of liquid or other matter. This theory is applicable not only to the processes mentioned but also to the occurrence of an electric breakdown in a dielectric, the processes of gel formation when the concentration of a substance or temperature changes, and many others [28,29,30].

The main qualitative characteristic of the lattice, which simulates the environment, is the ability to “transmit,” i.e., ensure the “flow.” As a rule, infinite lattices are considered in modeling, i.e., containing unlimited nodes and links. The formation of percolation on a cluster describes the possibility of “percolation” in the medium (sometimes called an infinite cluster). When modeling the lattice, it is assumed that the lattice cells, i.e., nodes or links between them, can be of two types: the first capable of performing the function of “flow” (white cells), i.e., skip and the second-not capable of performing this function (black cells).

A parameter is used to characterize this ability numerically, referred to as the percolation threshold 0 < p_c < 1, which is numerically equal to the fraction of cells capable of performing the “leakage” function.

For a lattice of finite size, p_c is random, and its values are in the interval δ. However, as the number of cells, N, increases in the lattice, the interval δ narrows.

$$\delta \left(N\right)\approx \frac{c}{\sqrt[v]{N}}$$

(15)

where c—constant (c ≈ 0.5), and v—critical index (index of the correlation radius) depending on the dimension of the task (for a three-dimensional task v ≈ 0.8…0.9).

$$p_c=\underset{N\to \infty }{lim}\left(p_c\left(N\right)\right)$$

(16)

The value p_c is the threshold value of the fraction of “conducting” sites, corresponding to the phase transition state of the considered lattice (medium) from the non-conducting to the conducting state. The p_c value is different for different types of gratings.

A very small change pc characterizes the transition of a medium from a non-conducting to a conducting state. A phase transition occurs when the fraction of conducting cells approaches p_c and takes place on a sufficiently small interval of variation of the value p_c. The medium is impermeable at values less than p_c, and at values exceeding p_c, it is permeable.

As you can see from the description of this model, it is formal enough to be used for a wide range of applications.

In the above model of the Internet of Things network located in a multi-story building, the network is also represented in the form of a lattice. These nodes, i.e., nodes of the communication network (possibly groups of nodes), are located in various rooms of the multi-story building. Connections between nodes are channels that can be formed between neighboring network nodes.

In a real network, such connections can exist with neighboring nodes; however, as seen from the previously introduced analysis of the channel quality, based on the signal-to-noise ratio, connections with neighboring nodes allow obtaining the highest channel quality, thus, ensuring the highest data transfer rate. The proposed network model in the form of a lattice can be considered a logical structure for organizing connections between nodes located in separate rooms, i.e., the structure of the lattice is determined by the structure of the structure.

If there is a functioning network with a communication node in the room, then the corresponding cell of the model is considered “white”, i.e., conductive. Otherwise, it is “black”, i.e., a non-conductive cell. We assume that connections are established only between neighboring nodes, i.e., located in adjacent premises. Then, a connected network in a building must have a percolation cluster of adjacent “white” cells.

Exact analytical solutions are known for only a small number of cases of two-dimensional (plane) lattices [26]. Simulation methods are used to analyze the behavior of other types of lattices. In particular, the percolation threshold for a simple cubic lattice used to model the network is p_c = 0.31 [29,30].

This value can be interpreted as the minimum share of premises in a multi-story building for the model under consideration. Functioning network nodes (white cells) should be located where it is possible to transfer messages between all network nodes, i.e., network connectivity is ensured. Percolation thresholds for some types of three-dimensional gratings are given in

Table 1. Percolation thresholds for some types of three-dimensional gratings.

	Lattice Type	Percolation Threshold
1	Simple cubic	0.310
2	Volumetric centered cubic	0.243
3	Face-centered cubic	0.195
4	Hexagonal	0.200

[26].

A simple cubic lattice assumes the presence of nodes at the vertices of the cube, a body-centered cubic lattice assumes the presence of another node in the center of the cell (cube), and a face-centered cubic lattice assumes the presence of nodes at the centers of each of the cube faces.

Most likely, cubic lattice models are applicable to modeling a network inside multi-story buildings. Along with the percolation threshold, other characteristics can be used to describe the network structure.

Correlation length, ξ, is the distance at which there is an order or characteristic size of a cluster of blocked cells for p < p_c or voids for p > p_c. It can be calculated as follows.

$$\xi \left(p\right)={\left|p-p_c\right|}^{-v}$$

(17)

where p—the proportion of white cells, p_c—percolation threshold, and v—critical index (correlation radius index).

When the area size is smaller than the correlation length ξ, a cluster of self-similar fractal dimensions is defined as follows.

$$d_f=d-\beta /v$$

(18)

where d—dimension of the task, β—method parameter (for 3D task β ≈ 0.4), and v—correlation radius index.

The percolation cluster possesses self-similarity; near the percolation defect, the fractal dimension of the percolation cluster is d_f = 2.54 [20].

The above parameters can be used to assess the quality of the Internet of Things network’s functioning in a multi-story building described by the cubic lattice model.

The correlation length can be used to indicate the number of nodes inaccessible in the network, and the fractal dimension of the cluster allows one to characterize the shape of the structures formed by the model cells (network nodes).

Thus, percolation theory’s provisions describe a three-dimensional network model in a multi-story building or similar structures. It should be noted that the provisions of this theory are also suitable for constructing two-dimensional models. The main parameters determined for the percolation structure can also be used to describe the properties of the communication network.

6. Conclusions

Developing a three-dimensional model of the IoT networks has shown that a lattice can describe its structure when building a network in a multi-story building. In most cases, these structures are close to a simple cubic lattice. In addition, analysis of the propagation of radio signals, taking into account obstacles in a multi-story building, showed that the signal-to-noise ratio at the points of location of nodes significantly depends on the power level of in-channel interference created by all nodes of the network. This determines the dependence on the number of nodes and the intensity of the generated traffic on the activity. Furthermore, the quality of the channel, determined by the signal-to-noise ratio at the points of location of the network nodes in the proposed model with typical parameters of the building structure, allows for most standards to provide the highest transmission speed. Moreover, the percolation theory provisions are applicable to the proposed model, which allows describing the network in terms of its connectivity, i.e., the ability to deliver messages between nodes and obtain additional characteristics of its structure. The proposed three-dimensional model and methods for assessing the quality of the functioning of the IoT networks can be adapted for other conditions by choosing the appropriate parameters of the model and changing the structure of connections.

Further work in this direction will focus on developing new types of lattices applicable to the problems of building communication networks and developing a simulation model for obtaining network characteristics and researching their properties based on the theory of percolation.